Koopman-based control of nonlinear systems with closed-loop guarantees

Robin Strässer; Julian Berberich; Manuel Schaller; Karl Worthmann; Frank Allgöwer

doi:10.1515/auto-2024-0162

Article

Koopman-based control of nonlinear systems with closed-loop guarantees

Robin Strässer
Robin Strässer received a master’s degree in Simulation Technology from the University of Stuttgart, Germany, in 2020. Since 2020, he has been a Research and Teaching Assistant with the Institute for Systems Theory and Automatic Control and a member of the Graduate School Simulation Technology at the University of Stuttgart. His research interests include data-driven system analysis and control, with a focus on nonlinear systems. Robin Strässer received the Best Poster Award at the International Conference on Data-Integrated Simulation Science (SimTech2023).
, Julian Berberich
Julian Berberich is a Lecturer (Akademischer Rat) at the Institute for Systems Theory and Automatic Control at the University of Stuttgart, Germany. He received his Ph.D. in Mechanical Engineering in 2022, and a Master’s degree in Engineering Cybernetics in 2018, both from the University of Stuttgart, Germany. In 2022, he was a visiting researcher at ETH Zürich, Switzerland. He is a recipient of the 2022 George S. Axelby Outstanding Paper Award as well as the Outstanding Student Paper Award at the 59th IEEE Conference on Decision and Control in 2020. His research interests include data-driven analysis and control as well as quantum computing.
, Manuel Schaller
Manuel Schaller obtained the M.Sc. and Ph.D. in Applied Mathematics from the University of Bayreuth in 2017 and 2021 respectively. From 2020 to 2023 he held a Lecturer position and a Junior Professorship at Technische Universität Ilmenau, Germany. Since August 2024, he is tenure track assistant professor at Chemnitz University of Technology, Germany. His research focuses on data-driven control with guarantees, port-Hamiltonian systems and efficient numerical methods for optimal control. For his research he has been named junior fellow of the GAMM (Society for Applied Mathematics and Mechanics), received the Best Poster Award at the workshop on systems theory and PDEs (WOSTAP 2022) and is an elected member of the Young Academy of the European Mathematical Society (EMS) in 2024–2027.
, Karl Worthmann
Karl Worthmann received his Ph.D. degree in mathematics from the University of Bayreuth, Germany, in 2012. 2014 he become assistant professor for “Differential Equations” at Technische Universität Ilmenau (TU Ilmenau), Germany. 2019 he was promoted to full professor after receiving the Heisenberg-professorship “Optimization-based Control” by the German Research Foundation (DFG). He was recipient of the Ph.D. Award from the City of Bayreuth, Germany, and stipend of the German National Academic Foundation. 2013 he has been appointed Junior Fellow of the Society of Applied Mathematics and Mechanics (GAMM), where he served as speaker in 2014 and 2015. His current research interests include systems and control theory with a particular focus on nonlinear model predictive control, stability analysis, and data-driven control.
and Frank Allgöwer
Frank Allgöwer studied engineering cybernetics and applied mathematics in Stuttgart and with the University of California, Los Angeles (UCLA), CA, USA, respectively, and received the Ph.D. degree from the University of Stuttgart, Stuttgart, Germany. Since 1999, he has been the Director of the Institute for Systems Theory and Automatic Control and a professor with the University of Stuttgart. His research interests include predictive control, data-based control, networked control, cooperative control, and nonlinear control with application to a wide range of fields including systems biology. Dr. Allgöwer was the President of the International Federation of Automatic Control (IFAC) in 2017–2020 and the Vice President of the German Research Foundation DFG in 2012–2020.

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From the journal at - Automatisierungstechnik Volume 73 Issue 6

Abstract

In this paper, we provide a tutorial overview and an extension of a recently developed framework for data-driven control of unknown nonlinear systems with rigorous closed-loop guarantees. The proposed approach relies on the Koopman operator representation of the nonlinear system, for which a bilinear surrogate model is estimated based on data. In contrast to existing Koopman-based estimation procedures, we state guaranteed bounds on the approximation error using the stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD) framework. The resulting surrogate model and the uncertainty bounds allow us to design controllers via robust control theory and sum-of-squares optimization, guaranteeing desirable properties for the closed-loop system. We present results on stabilization both in discrete and continuous time, and we derive a method for controller design with performance objectives. The benefits of the presented framework over established approaches are demonstrated with a numerical example.

Zusammenfassung

In dieser Arbeit geben wir eine Übersicht und Erweiterung eines kürzlich entwickelten Frameworks zur datengetriebenen Regelung unbekannter nichtlinearer Systeme mit rigorosen Garantien. Der betrachtete Ansatz basiert auf der Koopman-Operator-Darstellung des nichtlinearen Systems, für das ein bilineares Ersatzmodell anhand von Daten geschätzt wird. Im Gegensatz zu bestehenden Verfahren zur Koopman-basierten Schätzung bieten die gezeigten Methoden garantierte Schranken auf den Approximationsfehler unter Verwendung des stability- and certificate-oriented extended dynamic mode decomposition (SafEDMD)-Frameworks. Das resultierende Ersatzmodell und die Unsicherheitsgrenzen ermöglichen die Auslegung von Reglern mittels robuster Regelungstheorie und Sum-of-Squares-Optimierung und gewährleisten dabei wünschenswerte Eigenschaften für das geregelte System. Wir präsentieren Ergebnisse zur Stabilisierung sowohl in diskreter als auch in kontinuierlicher Zeit und entwickeln eine Methode zum Reglerentwurf mit Performancekriterien. Die Vorteile des vorgestellten Frameworks gegenüber etablierten Ansätzen werden anhand eines numerischen Beispiels demonstriert.

Keywords: data-driven control; Koopman theory; EDMD; nonlinear systems

Schlagwörter: datenbasierte Regelung; Koopman-Theorie; EDMD; nichtlineare Systeme

Corresponding author: Robin Strässer, Institute for Systems Theory and Automatic Control, University of Stuttgart, 70550 Stuttgart, Germany, E-mail: robin.straesser@ist.uni-stuttgart.de

About the authors

Robin Strässer

Robin Strässer received a master’s degree in Simulation Technology from the University of Stuttgart, Germany, in 2020. Since 2020, he has been a Research and Teaching Assistant with the Institute for Systems Theory and Automatic Control and a member of the Graduate School Simulation Technology at the University of Stuttgart. His research interests include data-driven system analysis and control, with a focus on nonlinear systems. Robin Strässer received the Best Poster Award at the International Conference on Data-Integrated Simulation Science (SimTech2023).

Julian Berberich

Julian Berberich is a Lecturer (Akademischer Rat) at the Institute for Systems Theory and Automatic Control at the University of Stuttgart, Germany. He received his Ph.D. in Mechanical Engineering in 2022, and a Master’s degree in Engineering Cybernetics in 2018, both from the University of Stuttgart, Germany. In 2022, he was a visiting researcher at ETH Zürich, Switzerland. He is a recipient of the 2022 George S. Axelby Outstanding Paper Award as well as the Outstanding Student Paper Award at the 59th IEEE Conference on Decision and Control in 2020. His research interests include data-driven analysis and control as well as quantum computing.

Manuel Schaller

Manuel Schaller obtained the M.Sc. and Ph.D. in Applied Mathematics from the University of Bayreuth in 2017 and 2021 respectively. From 2020 to 2023 he held a Lecturer position and a Junior Professorship at Technische Universität Ilmenau, Germany. Since August 2024, he is tenure track assistant professor at Chemnitz University of Technology, Germany. His research focuses on data-driven control with guarantees, port-Hamiltonian systems and efficient numerical methods for optimal control. For his research he has been named junior fellow of the GAMM (Society for Applied Mathematics and Mechanics), received the Best Poster Award at the workshop on systems theory and PDEs (WOSTAP 2022) and is an elected member of the Young Academy of the European Mathematical Society (EMS) in 2024–2027.

Karl Worthmann

Karl Worthmann received his Ph.D. degree in mathematics from the University of Bayreuth, Germany, in 2012. 2014 he become assistant professor for “Differential Equations” at Technische Universität Ilmenau (TU Ilmenau), Germany. 2019 he was promoted to full professor after receiving the Heisenberg-professorship “Optimization-based Control” by the German Research Foundation (DFG). He was recipient of the Ph.D. Award from the City of Bayreuth, Germany, and stipend of the German National Academic Foundation. 2013 he has been appointed Junior Fellow of the Society of Applied Mathematics and Mechanics (GAMM), where he served as speaker in 2014 and 2015. His current research interests include systems and control theory with a particular focus on nonlinear model predictive control, stability analysis, and data-driven control.

Frank Allgöwer

Frank Allgöwer studied engineering cybernetics and applied mathematics in Stuttgart and with the University of California, Los Angeles (UCLA), CA, USA, respectively, and received the Ph.D. degree from the University of Stuttgart, Stuttgart, Germany. Since 1999, he has been the Director of the Institute for Systems Theory and Automatic Control and a professor with the University of Stuttgart. His research interests include predictive control, data-based control, networked control, cooperative control, and nonlinear control with application to a wide range of fields including systems biology. Dr. Allgöwer was the President of the International Federation of Automatic Control (IFAC) in 2017–2020 and the Vice President of the German Research Foundation DFG in 2012–2020.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: The authors state no conflict of interest.
Research funding: F. Allgöwer is thankful that this work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy – EXC 2075 – 390740016 and within grant AL 316/15-1 – 468094890. K. Worthmann gratefully acknowledges funding by the German Research Foundation (DFG; grant WO 2056/14-1 – 507037103). R. Strässer thanks the Graduate Academy of the SC SimTech for its support.
Data availability: Not applicable.

Appendix A Proof of Theorem 3

We consider the Lyapunov function candidate V ( x ) = Φ ̂ ( x ) ⊤ P − 1 Φ ̂ ( x ) for some positive definite P⁻¹ ≻ 0, where a suitable lower and upper bound on V(x) can be constructed using (6); see [30], Eq. (46)]. In the following, we establish the Lyapunov inequality

(38) V ̇ ( x ) = d d t Φ ̂ ( x ) ⊤ P − 1 Φ ̂ ( x ) + Φ ̂ ( x ) ⊤ P − 1 d d t Φ ̂ ( x ) ≤ − ε ‖ x ‖ 2

with some ɛ > 0 for all x ∈ Ω(c*). To this end, we establish closed-loop stability of the bilinear surrogate with state Φ ̂ ( x ) before exploiting (6) to ensure that all trajectories of the original state x decay to zero if Φ ̂ ( x ) decays to zero. First, we define K(z) = L(z)P⁻¹ and

(39) A K ( z ) = A + B 0 K ( z ) + B ̃ ( K ( z ) ⊗ z ) .

Then, by exploiting L(z) ⊗ z = (K(z) ⊗ z)P, Q(z) ∈ SOS[z,2α]^2N+m ensures

(40) A K ( z ) P + P A K ( z ) ⊤ + ρ I N + τ ( z ) I N P P K ( z ) ⊤ P − τ ( z ) 2 c x 2 I N 0 K ( z ) P 0 − τ ( z ) 2 c u 2 I m ⪯ 0

for all z ∈ R N .^[4] Multiplying from the left and the right by blkdiag(P⁻¹, I, I) leads to

(41) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 + τ ( z ) P − 2 I N K ( z ) ⊤ I N − τ ( z ) 2 c x 2 I N 0 K ( z ) 0 − τ ( z ) 2 c u 2 I m ⪯ 0

which is equivalent to

(42) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 I N K ( z ) ⊤ I N − τ ( z ) 2 c x 2 I N 0 K ( z ) 0 − τ ( z ) 2 c u 2 I m − P − 1 0 0 − τ ( z ) I N P − 1 0 0 ⊤ ⪯ 0

for all z ∈ R n . Now, we apply the Schur complement and reorder the block rows and columns to obtain equivalently

(43) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 P − 1 I N K ( z ) ⊤ P − 1 − 1 τ ( z ) I N 0 0 I N 0 − τ ( z ) 2 c x 2 I N 0 K ( z ) 0 0 − τ ( z ) 2 c u 2 I m ⪯ 0 .

Using again the Schur complement, now w.r.t. the last two block rows and columns, (43) reads equivalently

(44) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 P − 1 P − 1 − 1 τ ( z ) I N + ⋆ ⊤ 2 c x 2 τ ( z ) I N 0 0 2 c u 2 τ ( z ) I m I 0 K ( z ) 0 ⪯ 0 .

Note that (44) can be decomposed into

(45) ⋆ ⊤ ρ P − 2 P − 1 P − 1 0 I 0 A K ( z ) I + 1 τ ( z ) ⋆ ⊤ 2 c x 2 I N 0 0 0 2 c u 2 I m 0 0 0 − I N I 0 K ( z ) 0 0 I ⪯ 0 .

Substituting z by Φ ̂ ( x ) ∈ R N , multiplying (45) from the left and the right by Φ ̂ ( x ) ⊤ r ( x , u ) ⊤ and its transpose, respectively, and applying the generalized S-procedure [47], [48] with multiplier τ ( Φ ̂ ( x ) ) − 1 , we deduce

(46) ⋆ ⊤ ρ P − 2 P − 1 P − 1 0 I 0 A K ( Φ ̂ ( x ) ) I Φ ̂ ( x ) r ( x , u ) ≤ 0

for all x ∈ R n if the residual r satisfies

(47) ⋆ ⊤ 2 c x 2 I N 0 0 0 2 c u 2 I m 0 0 0 − I N I 0 K ( Φ ̂ ( x ) ) 0 0 I Φ ̂ ( x ) r ( x , u ) ≥ 0 .

Now, recall d d t Φ ̂ ( x ) = A K ( Φ ̂ ( x ) ) Φ ̂ ( x ) + r ( x , u ) to observe that (46) is equivalent to

(48) V ̇ ( x ) ≤ − ρ Φ ̂ ( x ) ⊤ P − 2 Φ ̂ ( x ) ≤ − ρ σ min ( P − 1 ) 2 ‖ Φ ̂ ( x ) ‖ 2 ≤ − ρ ‖ P ‖ 2 2 L Φ 2 ‖ x ‖ 2 ,

where we exploit (6). Hence, we have established the desired Lyapunov inequality (41) with ε = ρ ‖ P ‖ 2 2 L Φ 2 for all r satisfying (47). Since all r satisfying the residual bound (21) also satisfy

(49) ‖ r ( x , u ) ‖ 2 ≤ 2 c x 2 ‖ Φ ̂ ( x ) ‖ 2 + 2 c u 2 ‖ u ‖ 2 ,

the residual satisfies (47) for u = K ( Φ ̂ ( x ) ) Φ ̂ ( x ) . It remains to show positive invariance of the set Ω(c*). In particular, the proportional error bound (21) is only guaranteed to hold on X with probability 1 − δ. Hence, guarantees for the nonlinear system can only be deduced if the closed-loop system remains in X for all times. To this end, we define the Lyapunov sublevel set Ω(c) in (23a) and maximize the value c according to (23b) such that Ω(c*) is the maximal Lyapunov sublevel set in X . Then, Ω(c*) characterizes the largest guaranteed region of attraction in the sampling region X , which guarantees closed-loop exponential stability of the nonlinear system (1) with probability 1 − δ.□

Appendix B Proof of Theorem 7

In the following, we establish exponential stability and quadratic performance of the lifted system (30) and thus, if the proportional error bound (21) holds, of the nonlinear system (26). To this end, we define the Lyapunov candidate function V ( x ) = Φ ̂ ( x ) ⊤ P − 1 Φ ̂ ( x ) and recall A_K(z) in (39). Further, we define K(z) = L(z)P⁻¹ and

(50) C K ( z ) = C + D K ( z ) + D ̃ ( K ( z ) ⊗ z ) ,

(51) H w = Q w + S w D w + D w ⊤ S w ⊤ .

Then, multiplying (34) from the left and the right by

(52) P − 1 0 0 0 0 0 0 0 1 λ ( z ) I N 0 0 I N 0 0 0 0 0 I m 0 0 0 0 0 0 I q

and its transpose, respectively, yields

(53) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 ⋆ ⋆ ⋆ ⋆ B w ( z ) ⊤ P − 1 + 1 λ ( z ) S w C K ( z ) 1 λ ( z ) H w + η λ ( z ) 2 I p ⋆ ⋆ ⋆ I 0 − τ ( z ) 2 c x 2 I N ⋆ ⋆ K ( z ) 0 0 − τ ( z ) 2 c u 2 I m ⋆ C K ( z ) D w 0 0 − λ ( z ) R w − 1 − P − 1 0 0 0 0 ( − τ ( z ) I N ) ⋆ ⊤ ⪯ 0

for all z ∈ R N . We apply the Schur complement and reorder the block rows and columns to obtain equivalently

(54) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 ⋆ ⋆ ⋆ ⋆ ⋆ P − 1 − 1 τ ( z ) I N ⋆ ⋆ ⋆ ⋆ B w ( z ) ⊤ P − 1 + 1 λ ( z ) S w C K ( z ) 0 1 λ ( z ) H w + η λ ( z ) 2 I p ⋆ ⋆ ⋆ I 0 0 − τ ( z ) 2 c x 2 I N ⋆ ⋆ K ( z ) 0 0 0 − τ ( z ) 2 c u 2 I m ⋆ C K ( z ) 0 D w 0 0 − λ ( z ) R w − 1 ⪯ 0 .

Using again the Schur complement, now w.r.t. the three last block rows and columns, (54) is equivalent to

(55) P − 1 A K ( z ) + A K ( z ) ⊤ P − 1 + ρ P − 2 ⋆ ⋆ P − 1 − 1 τ ( z ) I N ⋆ B w ( z ) ⊤ P − 1 + 1 λ ( z ) S w C K ( z ) 0 1 λ ( z ) H w + η λ ( z ) 2 + ⋆ ⊤ 2 c x 2 τ ( z ) I N 0 0 0 2 c u 2 τ ( z ) I m 0 0 0 1 λ ( z ) R w I 0 0 K ( z ) 0 0 C K ( z ) 0 D w ⪯ 0

Note that (55) can be equivalently decomposed into

(56) ⋆ ⊤ ρ P − 2 P − 1 P − 1 0 I 0 0 A K ( z ) I B w ( z ) + 1 τ ( z ) ⋆ ⊤ 2 c x 2 I N 0 0 0 2 c u 2 I m 0 0 0 − I N I 0 0 K ( z ) 0 0 0 I 0 + 1 λ ( z ) ⋆ ⊤ Q w + η λ ( z ) S w S w ⊤ R w 0 0 I C K ( z ) 0 D w ⪯ 0 .

Recall (32) for the supply rate s(w, y) and (30) for the lifted (closed-loop) dynamics, i.e.,

(57) d d t Φ ̂ ( x ) = A K ( Φ ̂ ( x ) ) Φ ̂ ( x ) + r ( x , u ) + B w ( Φ ̂ ( x ) ) w ,

(58) y = C K ( Φ ̂ ( x ) ) Φ ̂ ( x ) + D w w .

Then, substituting z by Φ ̂ ∈ R N , multiplying (56) from the left and the right by Φ ̂ ( x ) ⊤ r ( x , u ) ⊤ w ⊤ and its transpose, respectively, and applying the generalized S-procedure [47] with multiplier τ ( Φ ̂ ( x ) ) − 1 yields

(59) V ̇ ( x ) ≤ − ρ ‖ P ‖ 2 2 ‖ Φ ̂ ( x ) ‖ 2 − η λ ( Φ ̂ ( x ) ) 2 ‖ w ‖ 2 − 1 λ ( Φ ̂ ( x ) ) s ( w , y )

for all x ∈ R n and r satisfying the proportional bound (21) according to the discussion around (49). Here, we have used (6) to bound the norm of Φ ̂ ( x ) in terms of x. Moreover, we define λ * = min z ∈ R N λ ( z ) , where we know λ* > 0 since λ ∈ SOS₊[z, 2β]. Thus, we deduce for ɛ = η λ * the inequality

(60) V ̇ ( x ) ≤ − ε λ ( z ) ‖ w ‖ 2 − 1 λ ( z ) s ( w , y ) .

Hence, by integrating (60) from t = 0 to ∞, we establish (33) by using x(0) = 0 and lim_t→∞x(t) = 0, and, thus, quadratic performance according to Definition 6.□

Appendix C Proof of Corollary 9

Recall that Theorem 7 establishes the Lyapunov inequality (59) for all x ∈ X , w ∈ L 2 and, thus, for all x ∈ Ω ( c * ) ⊆ X , ‖w‖² ≤ ν. To show robust positive invariance of Ω(c*), we need to construct ν such that x(t + χ) ∈ Ω(c*) for all x(t) ∈ Ω(c*), ‖w‖² ≤ ν, and χ > 0. The established Lyapunov inequality (59) leads to

(61) V ̇ ( x ) ≤ − ρ ‖ P ‖ 2 2 ‖ Φ ̂ ( x ) ‖ 2 + 1 λ ( Φ ̂ ( x ) ) α ( ‖ w ‖ 2 ) .

Using V ( x ) ≤ ‖ Φ ̂ ( x ) ‖ 2 ‖ P ‖ 2 − 1 or, in particular, ‖ Φ ̂ ( x ) ‖ 2 ≥ V ( x ) ‖ P ‖ 2 , yields

(62) V ̇ ( x ( t ) ) ≤ − β V ( x ( t ) ) + 1 λ ( Φ ̂ ( x ( t ) ) ) α ( ‖ w ( t ) ‖ 2 ) ,

where we define β = ρ ‖ P ‖ 2 3 . Then, we multiply both sides of (62) by e^βt and use ‖w(t)‖² ≤ ν≔α⁻¹(λ*c*) to obtain

(63) e β t d d t V ( x ( t ) ) + β e β t V ( x ( t ) ) ≤ c * e β t .

Note that the left-hand side of the inequality is just the total time derivative of e^βtV(x(t)). Thus, integrating both sides on the interval [t, t + χ] for any χ > 0 yields

(64) e β ( t + χ ) V ( x ( t + χ ) ) − e β t V ( x ( t ) ) ≤ c * ∫ t t + χ e β s d s = c * e β t e β χ − 1 .

Hence,

(65) V ( x ( t + χ ) ) ≤ e − β χ V ( x ( t ) ) + c * 1 − e − β χ .

For all x(t) ∈ Ω(c*), i.e., V(x(t)) ≤ c*, we obtain

(66) V ( x ( t + χ ) ) ≤ e − β χ c * + c * 1 − e − β χ = c *

for all ‖w(t)‖² ≤ ν. Thus, we have established the existence of a ν > 0 such that the RoA Ω(c*) is robust positively invariant.

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Received: 2024-11-15

Accepted: 2025-03-04

Published Online: 2025-05-28

Published in Print: 2025-06-26

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Articles in the same Issue

https://doi.org/10.1515/auto-2024-0162

Keywords for this article

data-driven control; Koopman theory; EDMD; nonlinear systems